Article Dans Une Revue The Annals of Applied Probability Année : 2022

Interval fragmentations with choice: equidistribution and the evolution of tagged fragments

Résumé

We consider a Markovian evolution on point processes, the $\Psi$--process, on the unit interval in which points are added according to a rule that depends only on the spacings of the existing point configuration. Having chosen a spacing, a new point is added uniformly within it. Building on previous work of the authors and of Junge, we show that the empirical distribution of points in such a process is always equidistributed under mild assumptions on the rule, generalizing work of Junge. A major portion of this article is devoted to the study of a particular growth--fragmentation process, or cell process, which is a type of piecewise--deterministic Markov process (PDMP). This process represents a linearized version of a size--biased sampling from the $\Psi$--process. We show that this PDMP is ergodic and develop the semigroup theory of it, to show that it describes a linearized version of the $\Psi$--process. This PDMP has appeared in other contexts, and in some sense we develop its theory under minimal assumptions.

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hal-03292333 , version 1 (07-07-2025)

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Pascal Maillard, Elliot Paquette. Interval fragmentations with choice: equidistribution and the evolution of tagged fragments. The Annals of Applied Probability, 2022, 32 (5), ⟨10.1214/21-AAP1766⟩. ⟨hal-03292333⟩
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